sheaf
1 Presheaves
Let $X$ be a topological space^{} and let $\mathcal{A}$ be a category^{}. A presheaf^{} on $X$ with values in $\mathcal{A}$ is a contravariant functor^{} $F$ from the category $\mathcal{B}$ whose objects are open sets in $X$ and whose morphisms^{} are inclusion mappings of open sets of $X$, to the category $\mathcal{A}$.
As this definition may be less than helpful to many readers, we offer the following equivalent^{} (but longer) definition. A presheaf $F$ on $X$ consists of the following data:

1.
An object $F(U)$ in $\mathcal{A}$, for each open set $U\subset X$

2.
A morphism ${\mathrm{res}}_{V,U}:F(V)\to F(U)$ for each pair of open sets $U\subset V$ in $X$ (called the restriction^{} morphism), such that:

(a)
For every open set $U\subset X$, the morphism ${\mathrm{res}}_{U,U}$ is the identity morphism.

(b)
For any open sets $U\subset V\subset W$ in $X$, the diagram

(a)